A335453 Number of (2,1,2)-matching permutations of the prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jun 14 2020

Keywords

• nonn

Comments

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Examples

			The a(n) permutations for n = 18, 36, 54, 72, 90, 108, 144, 180:
  (212)  (1212)  (2122)  (11212)  (2123)  (12122)  (111212)  (12123)
         (2112)  (2212)  (12112)  (2132)  (12212)  (112112)  (12132)
         (2121)          (12121)  (2312)  (21122)  (112121)  (12312)
                         (21112)  (3212)  (21212)  (121112)  (13212)
                         (21121)          (21221)  (121121)  (21123)
                         (21211)          (22112)  (121211)  (21132)
                                          (22121)  (211112)  (21213)
                                                   (211121)  (21231)
                                                   (211211)  (21312)
                                                   (212111)  (21321)
                                                             (23112)
                                                             (23121)
                                                             (31212)
                                                             (32112)
                                                             (32121)
		

Links

• Wikipedia, Permutation pattern
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

References found in the link are not all repeated here.

Positions of ones are A095990.

The avoiding version is A335450.

Replacing (2,1,2) with (1,2,1) gives A335446.

Patterns are counted by A000670.

Permutations of prime indices are counted by A008480.

Unsorted prime signature is A124010. Sorted prime signature is A118914.

(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.

STC-numbers of permutations of prime indices are A333221.

(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.

Patterns matched by standard compositions are counted by A335454.

(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.

(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.

Dimensions of downsets of standard compositions are A335465.

(1,2,2)-matching compositions are ranked by A335475.

(2,2,1)-matching compositions are ranked by A335477.

Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

Formula

a(n) + A335450(n) = A008480(n).